Bounds on the Hyperbolicity Constant

If X is a geodesic metric space and x1,x2,x3∈X, a geodesic triangle T={x1,x2,x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic in the Gromov sense if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by δ(X) the sharp hyperbolicity constant of X, i.e. . To compute the hyperbolicity constant is a very hard problem. Then it is natural to try to bound the hyperbolycity constant in terms of some parameters of the graph. Denote by G(n,m) the set of graphs G with n vertices and m edges, and such that every edge has length 1. In this work we estimate A(n,m):=min⁡{δ(G)|G∈G(n,m)} and B(n,m):=max⁡{δ(G)|G∈G(n,m)}. In particular, we obtain good bounds for A(n,m) and B(n,m), and we compute the precise value of A(n,m) for many values of n and m. In addition, we obtain an upper bound of the size of any graph in terms of its diameter and its order.